Invariant SubspacesThis broad survey spans a wealth of studies on invariant subspaces, focusing on operators on separable Hilbert space. Largely self-contained, it requires only a working knowledge of measure theory, complex analysis, and elementary functional analysis. Subjects include normal operators, analytic functions of operators, shift operators, examples of invariant subspace lattices, compact operators, and the existence of invariant and hyperinvariant subspaces. Additional chapters cover certain results on von Neumann algebras, transitive operator algebras, algebras associated with invariant subspaces, and a selection of unsolved problems. 1973 edition. New appendix on recent developments. |
Contents
Introduction and Preliminaries | 1 |
Shift Operators | 22 |
Examples of Invariant Subspace Lattices | 60 |
Compact Operators | 84 |
Existence of Invariant and Hyperinvariant Subspaces | 95 |
Certain Results on von Neumann Algebras | 117 |
Transitive Operator Algebras | 138 |
Algebras Associated with Invariant Subspaces | 167 |
Some Unsolved Problems | 192 |
References | 199 |
List of Symbols | 211 |
Some Subsequent Developments | 221 |
Additional References | 231 |
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Common terms and phrases
algebra containing Amer Arveson attainable lattices Banach space Borel set bounded closure commutes compact operator converges Corollary countable cyclic defined Definition denote dense direct sum Donoghue operator eigenvector equation finite rank finite-dimensional space follows Funct Halmos Hence Hermitian operator Hilbert space hyperreducible hypothesis identity implies inner function invariant linear manifold invariant sub invariant subspace problem Lemma linear transformation Lomonosov's M₁ M₂ Mathematical Neumann algebra non-trivial hyperinvariant subspace non-trivial invariant subspace non-zero Nordgren normal operators Note nullspace operator algebras Operator Theory orthogonal orthonormal basis partial isometry Pearcy polynomially compact positive integer Proc projection proof of Theorem Proposition quasinilpotent quasitriangular Radjavi range reducing subspaces reductive algebra reductive operator result Rosenthal self-adjoint shift of multiplicity spectral theorem subalgebra of B(H subset Suppose transitive algebra problem trivial U₁ unilateral shift unitarily equivalent unitary operator vector von Neumann algebra weakly closed algebra